Occasional Paper. Risk Measurement Illiquidity Distortions. Jiaqi Chen and Michael L. Tindall
|
|
- Sharon Edwards
- 5 years ago
- Views:
Transcription
1 DALLASFED Occasional Paper Risk Measurement Illiquidity Distortions Jiaqi Chen and Michael L. Tindall Federal Reserve Bank of Dallas Financial Industry Studies Department Occasional Paper 12-2 December 212
2 Risk measurement illiquidity distortions by Jiaqi Chen and Michael L. Tindall Financial Industry Studies Department Federal Reserve Bank of Dallas * November 212 Abstract We examine the effects of smoothed hedge fund returns on standard deviation, skewness, and kurtosis of return and on correlation of returns and cross-sectional volatility and covariance of returns using an MA(2)- GARCH(1,1)-skewed-t representation of returns instead of the traditional MA(2) model employed in the literature. We present evidence that our proposed representation is more consistent with the behavior of hedge fund returns and that the traditional method tends to overstate the degree of smoothing observed in hedge fund returns. We present methods for correcting for the distortive effects of smoothing using our representation. Keywords: hedge fund, return smoothing, illiquidity, GARCH * Please direct correspondence to michael.tindall@dal.frb.org. The views expressed herein are those of the authors and do not necessarily reflect the views of the Federal Reserve Bank of Dallas or the Federal Reserve System.
3 2 1. Introduction Hedge fund returns are characterized by certain stylized facts. Among them, hedge fund return distributions are often skewed and leptokurtic, returns may exhibit high positive correlation between funds during periods of financial distress, and monthly returns of individual hedge funds are often positively serially correlated, a phenomenon known to hedge fund analysts as smoothed returns. Several writers have examined return smoothing. These include Asness, Krail, and Liew (22); Getmansky, Lo, and Makarov (24); Aragon (27); Kosowski, Naik, and Teo (27); Jagannathan, Malakov, and Novikov (21); and Titman and Tiu (21). Getmansky et al argue in their well-known work that hedge fund monthly returns are smoothed because of the effects of various forms of illiquidity. For example, for its month-end books a hedge fund may average broker estimates of an illiquid asset s value. Brokers may estimate the month-end value as a markup at an assumed rate from the prior month s value. This and the practice of averaging the estimates produces smoothed returns. It is established in the literature that with smoothed returns, standard deviation using reported returns is understated so that the Sharpe ratio and information ratio are overstated. It is also established that measures of skewness and kurtosis are also distorted by smoothing as is correlation of returns between funds. We analyze corrected measures of these statistics. We also examine the possibility that smoothing distorts crosssectional measures of returns, indicators of hedge fund risk, and we present a method for analyzing this issue. Section 2 examines computational methods employed in our analysis. Section 3 examines the impact of smoothed hedge fund returns on measured standard deviation, skewness, and kurtosis of individual hedge fund return streams. Section 4 examines the impact of smoothing on correlation of returns between hedge funds, and cross-sectional measures of returns between hedge funds. Section 5 presents an empirical study of key statistics under return smoothing. Section 6 presents conclusions. 2. Computational methods In the literature an MA(2) model of monthly hedge fund returns is often employed as follows: (2.1) R t o = θ R t + θ 1 R t 1 + θ 2 R t 2
4 3 1 = θ + θ 1 + θ 2 θ i 1 where R t o is the observed, or reported, monthly return in period t, R t is the underlying actual return in t, and the θs are parameters. Some authors (Getmansky et al 24) propose to estimate the MA(2) smoothing model using maximum likelihood estimation (MLE) while relaxing the constraint θ i 1. They define the de-meaned observed return X t = R t o µ. Then: (2.2) X t = θ ε t + θ 1 ε t 1 + θ 2 ε t 2 1 = θ + θ 1 + θ 2 ϵ k ~ iid N(, σ 2 ) It then becomes a traditional MA time series estimation process and can be estimated using standard statistical software. A key assumption in such an approach is that the underlying true hedge fund return has an iid normal distribution. In reality, the true return rarely has such a distribution. It is not uncommon for the return to show volatility clustering and skewness. To demonstrate this, we employ a dataset of 256 hedge funds from the Hedge Fund Research Inc. database, each with a continuous monthly reporting history for the period from January 1998 to December 21. We restricted our selection of hedge funds to those with assets under management of at least $1 million and continuous reporting of returns over our sample period. In Figure 1 below, we show the residuals for one of the funds obtained through the MA(2) process with normality assumptions. The distribution of the residuals, which should be iid normal, exhibits fat tails and are highly skewed.
5 4 Figure 1. Histogram of the residuals for fund #3 in the MA(2) model 6 Histogram of residuals for fund #3 in MA(2) model Residuals (%) To demonstrate the volatility clustering or heteroscedasticity, we show the autocorrelation function plot of the absolute residuals in the MA(2) model in Figure 2.
6 5 Figure 2. ACF of the absolute residuals for Fund #3 in the MA(2) model. The dashed horizontal lines form the 95 percent confidence band for no autocorrelation. ACF of absolute residuals for fu ACF Lag To extract the true return accommodating its skewness, kurtosis and heteroscedasticity properties, we propose using an MA(2)-GARCH(1,1)-skewed-t model: (2.3) X t = θ a t + θ 1 a t 1 + θ 2 a t 2 1 = θ + θ 1 + θ 2 a t = σ t ϵ t σ t = α + α 1 a t 1 + β 1 σ t 1 where the error term ϵ t has a skewed-t distribution defined in Lambert and Laurent (21) as follows. Suppose x has a Student-t distribution with density g (.) and degrees of freedom υ. Potential skewness is introduced through the parameter ξ such that: Γ [( υ 1) / 2] υ 2 1 (2.4) E ( x ξ, υ) = ξ = m π Γ ( υ / 2) ξ
7 6 [ ] = = = + = s m y I m sy g s m y I m sy g s y f s m x y s m x V ) [,,) ( ) ( 1/ 2 ), ( 1 1 ), ( υ ξ υ ξ ξ ξ υ ξ ξ ξ υ ξ where ), ( υ ξ y f is the density. For our 256 hedge funds, 228 of them have returns different from a normal distribution at the.5 significance level based on Jarque and Bera (1987) statistic, which is asymptotically distributed as a chi-square random variable with two degrees of freedom. This is a strong indication that we should incorporate the skewed-t distribution in our estimation process. In our estimation we also relax the constraint of θ i 1 as in the literature. In the following table, we list the estimated θs using our modified model and the traditional MA(2) model together with the Jarque and Bera statistics for the duals ϵ t. We list only the first 1 funds ranked in increasing order of the estimated smoothing parameter θ from the traditional model to illustrate the difference between the two estimation processes.
8 7 Table 1. Comparison of traditional model and skewed-t model across hedge funds. MA(2)-GARCH(1,1)-skewed-t MA(2) θ θ 1 θ 2 θ θ 1 θ 2 J-B statistic From the table we see that there is significant amount of smoothing in hedge fund returns and that the normality assumption is violated as the Jarque and Bera statistic is 5.99 at the.5 significance level. When we compare the estimations of θ, the traditional process tends to overestimate the smoothing effects in hedge fund returns, and that could exaggerate the distortion of key hedge funds statistics when taking smoothing into consideration. The charts below show the comparison of θ estimations using the simple MA(2) model and our MA(2)-GARCH(1,1)-skewed-t model for our 256 funds. The kernel density was calculated using a normal kernel with plug-in optimal bandwidth choice.
9 8 Figure 3.1. Histogram comparison of θ estimations 8 θ histogram for MA(2) θ histogram for MA(2)-GARCH(1,1)-Skewed-t Figure 3.2. Kernel density comparison of θ estimations θ kernel density for hedge funds MA(2) MA(2)-GARCH(1,1)-Skewed-t
10 9 Figure 3.3. Scatter plot comparison of θ estimations 1.8 θ estimated from MA(2)-GARCH(1,1)-Skewed-t θ estimated from MA(2) In the chart above, the straight line shows where the two estimates are equal. It is evident from the graphs that the estimates for θ are quite different for these two models. The traditional model tends to overstate the smoothing factor based on our database. Our proposal is closer to reality in that it takes into consideration the skewness, kurtosis, and heteroscedasticity of the return distributions. 3. Smoothed returns and statistical moments of individual hedge funds Return smoothing and standard deviation of return Geltner (1991) shows that for (2.1) the standard deviation SD o of the reported returns R t o and the standard deviation SD of the actual returns R t are related as follows: (3.1) SD = ψ 1 SD o ψ 1 = 1/( θ 2 i ) 1/2 The term ψ 1 is greater than 1 where θ i < 1; i=,1,2, i.e., where returns are smoothed, so that the standard deviation of reported returns is less than the standard deviation of the actual underlying returns. The chart below shows ψ 1 plotted as functions of θ and θ 1 (θ 2 = 1 θ θ 1, of course).
11 1 Figure 4. Distortion factor ψ 1 under return smoothing ψ for standard deviation θ 1 θ The distortive effect of return smoothing, i.e., the value of ψ 1, is greatest where θ = θ 1 = θ 2 = 1/3. The understatement of standard deviation means that the Sharpe ratio and the information ratio computed from reported returns are higher than they would be using actual returns. For example, where θ = θ 1 = θ 2 = 1/3, ψ 1 = so that the standard deviation of the reported return is.5774 (= 1/1.732) that of the actual return, and the reported Sharpe ratio and the information ratio are each higher than their respective actual measures. Getmansky et al find that distortions of standard deviation are sometimes substantial in empirical hedge fund data. With smoothed returns, reported standard deviation is less than actual standard deviation, and the reported Sharpe ratio and information ratio are higher than their respective actual measures, and the degree of distortion is sensitive to the method of estimation. Return smoothing and higher statistical moments Higher moments are also distorted by smoothing (Cavenaile et al, 211). Let SK, SK o, K, and K o denote, respectively, skewness of actual returns, skewness of observed returns, excess kurtosis of actual returns, and excess kurtosis of observed returns, respectively. Then:
12 11 (3.2) SK = ψ 2 SK o ψ 2 = ( θ i 2 ) 3/2 /( θ i 3 ) (3.3) K = ψ 3 K o ψ 3 = ( θ 2 i ) 2 /( θ 4 i ) Of course, in the absence of return smoothing, ψ 2 = ψ 3 = 1, but these distortion factors are elevated where returns are smoothed and may be well above 1. The charts below show ψ 2 and ψ 3 plotted as functions of θ and θ 1. Figure 5. Distortion factor ψ 2 under return smoothing ψ 2 for skewness θ 1 θ
13 Figure 6. Distortion factor ψ 3 under return smoothing ψ 3 for excess kurtosis θ 1 θ Again, the distortive effects of return smoothing are greatest where θ = θ 1 = θ 2 = 1/3 for both skewness and excess kurtosis. With smoothed returns, actual skewness and actual excess kurtosis are greater than their respective measures taken from reported returns, and the distortion of kurtosis is greater than the distortion of skewness. 4. Smoothed returns and statistical measurements across hedge funds Correlation of returns under return smoothing There is evidence that, during periods of financial distress, hedge fund return streams become positively correlated with each other, and analysts use correlation of hedge fund returns as an indicator of potential financial distress. Unfortunately, the correct computation of correlation of returns may be undermined by return smoothing. Let x t o and y t o be the observed returns of two hedge funds, let x t and y t be the respective actual returns, and suppose again that the two return streams are subject to MA(2) smoothing so that:
14 13 4.1) x t o = θ x x t + θ x1 x t 1 + θ x2 x t 2 y t o = θ y y t + θ y1 y t 1 + θ y2 y t 2 θ xi = 1 θ yi = 1 Then, the correlation corr[x t, y t ] of the actual returns and the correlation corr[x t o, y t o ] of the observed returns are related as follows (Geltner, 1991): 4.2) corr[x t, y t ] = ψ 4 corr[x o t, y o t ] ψ 4 = ( θ 2 xi θ 2 yi ) 1/2 /( θ xi θ yi ) When the smoothing parameters are the same for both funds, i.e., where θ xi = θ yi ; i =,1,2, we have ψ 4 = 1. However, when the smoothing parameters are different, we have ψ 4 > 1. Figure 7 shows ψ 4 as a function of θ y and θ y1 where θ x = θ x1 = θ x2 = 1/3. Figure 7. Distortion factor ψ 4 under return smoothing ψ 4 for correlation θ y θ y In the case of identical smoothing parameters, actual correlation and observed correlation are equal, but correlation of actual returns is greater in absolute value than correlation using reported returns where the smoothing parameters are different.
15 14 Analysts may use correlation of returns as an indicator of the degree of distress in financial markets, and so correcting measured correlation, i.e., computing Ψ 4, may be a consideration in analyzing hedge fund return data. Cross-sectional volatility, covariance and correlation of returns under return smoothing Return smoothing can have effects on other statistical measures, and we round out our analysis with an examination of these effects. Analysts may use cross-sectional measures of returns as indicators of hedge fund risk. Adrian (27) defines cross-sectional volatility at a point in time as 1 N (R N i=1 i o ) 2 where R o i is the observed return of fund i and N is the number of hedge funds in the database. We note that there is not a term like ψ above that converts this indicator to 1 N (R N i=1 i) 2, i.e., cross-sectional volatility of actual returns. He defines cross-sectional covariance as R N 2 N i=1 t oi. Using Credit Suisse/ Tremont style indexes, he finds that increases in covariances tend to precede elevations in volatility. Cross-sectional correlation is the ratio of cross-sectional covariance to the square of cross-sectional volatility. Adrian finds no statistical evidence that increases in correlations precede rises in volatility. He finds that the Long Term Capital Management crisis of August 1998 was accompanied by a large negative covariance of returns. Using the same style indexes, we replicated his results, demonstrating that such measures are useful when working with hedge fund style indexes. 1 N N j=1,j i However, problems arise when we deal with individual hedge funds as we do in this study which may use very idiosyncratic strategies even within the same style classification. For instance, in our analysis using the definitions above, we are not able to detect significant negative cross-sectional covariance in our database of individual hedge funds during periods of financial crisis. We also note that, where a large constant is added to all the returns, cross-sectional volatility/covariance is larger while the true underlying performance differences across hedge funds remain the same. Adding a constant to the returns may be thought of as artificially creating a more bullish operating environment for hedge funds. With that in mind, we propose the following modified cross-sectional measures for our study: R t oj
16 15 Cross-sectional volatility = 1 N (R N i=1 i o u) 2 where u = 1 N R N i=1 i o Cross-sectional covariance = 1 N N (R N 2 N j=1,j i t oi u) (R oj i=1 t u) When taking return smoothing into consideration, we will use our approach to filter out the unsmoothed underlying true return and estimate the above mentioned crosssectional measures. 5. Empirical study of key statistics Using our original dataset of 256 hedge funds, we use both our MA(2)-GARCH(1,1)- skewed-t model and the MA(2) model to examine the distortion effects for standard deviation, skewness, excess kurtosis, and the cross-sectional measures considered above. In this empirical study, we relax the constraint of θ i 1 so that, for instance, standard deviation of reported returns may at times be less than that of actual returns. The following charts show the histogram comparisons of distortions for standard deviation, skewness, excess kurtosis and pair-wise correlations estimated from the two methods. The cross-sectional measures using the observed return and filtered return in our approach are shown from January 1998 to December 21. Figure 8. Histogram of ψ 1 4 Histogram for standard deviation distortion estimated from MA(2)-GARCH(1,1)-Skewed-t Histogram for standard deviation distortion estimated from MA(2)
17 16 Figure 9. Histogram of ψ 2 8 Histogram for skewness distortion estimated from MA(2)-GARCH(1,1)-Skewed-t 6 Histogram for skewness distortion estimated from MA(2) Figure 1. Histogram of ψ 3 8 Histogram for excess kurtosis distortion estimated from MA(2)-GARCH(1,1)-Skewed-t 6 Histogram for excess kurtosis distortion estimated from MA(2)
18 17 Figure 11. Histogram of ψ 4 15 Histogram for pairwise correlation distortion estimated from MA(2)-GARCH(1,1)-Skewed-t 15 Histogram for pairwise correlation distortion estimated from MA(2) Figure 12. Comparison of cross-sectional volatility 15 observed actual Cross-sectional Volatility 1 5 Jan98 Jan Jan2 Jan4 Jan6 Jan8 Jan1
19 18 Figure 13. Comparison of cross-sectional covariance -.1 Cross-sectional Covariance observed actual -.8 Jan98 Jan Jan2 Jan4 Jan6 Jan8 Jan1 Thus, in general using our method, standard deviation, skewness and excess kurtosis all show some distortions for most of the 256 funds in our study, while the distortions tend to be less than that estimated from the traditional method. The cross-sectional risk measures are affected to a lesser extent, and it would suffice to use only the observed return in this case. Most importantly, there are three big spikes in cross-sectional volatility and three big negative spikes in cross-sectional covariance corresponding to the Long Term Capital Management crisis in 1998, the internet bubble burst in 2, and the recent financial crisis in Conclusion It is established in the literature that hedge fund return smoothing causes distortions in hedge fund statistics. We propose a new MA(2)-GARCH(1,1)-skewed-t model to incorporate the skewness, kurtosis and heteroscedasticity effects often encountered in hedge fund returns. In addition, it has been reported in the literature that return smoothing causes standard deviation of reported returns to understate true volatility and that smoothing distorts correlation of returns. Skewness and excess kurtosis are understated using smoothed returns. In addition, we find that some statistics such as crosssectional volatility and covariance are not substantially distorted by illiquidity and so they may serve for risk-measurement purposes without correcting for illiquidity effects.
20 The construction of our indicators of hedge fund illiquidity, i.e., the ψ, is consistent with our analytical framework, and we suggest that such measures may be useful to policymakers who require correct assessment of risk in hedge fund space. 19
21 2 References Adrian, T., 27, Measuring Risk in the Hedge Fund Sector Federal Reserve Bank of New York, Current Issues in Economics and Finance, Volume 13, Number 3. Aragon, G., 27, Share restrictions and asset pricing: evidence from the hedge fund industry, Journal of Financial Economics, 83, Asness, C., R. Krail and J. Liew, 22, Do hedge funds hedge? Journal of Portfolio Management, 28, Cavenaile, L., A. Coën, and G. Hübner, 211, The impact of illiquidity and higher moments of hedge fund returns on their risk-adjusted performance and diversification potential, The Journal of Alternative Investments, Spring 211, vol. 13, no. 4, pp Geltner, D.M., 1991, Smoothing in appraisal-based returns, Journal of Real Estate Finance and Economics, 4, Getmansky, M., A.W. Lo and I. Makarov, 24, An econometric model of serial correlation and illiquidity in hedge fund returns, Journal of Financial Economics, 74, Jagannathan, R., A. Malakhov and D. Novikov, 21, Do hot hands exist among hedge fund managers? An empirical evaluation, Journal of Finance, 65, Jarque, C.M., and A.K. Bera, 1987, A test of normality of observations and regression residuals, International Statistical Review, 55, Kosowski, R., N.Y. Naik and M. Teo, 27, Do hedge funds deliver alpha? A Bayesian and bootstrap approach, Journal of Financial Economics, 84, Lambert, P. and S. Laurent, 21, Modeling financial time series using GARCH-type models and a skewed student density, Mimeo, Université de Liège. Lo, A.W., 21, Hedge funds, an analytic perspective, Princeton University Press. Titman, S. and C. Tiu, 211, Do the best hedge funds hedge? The Review of Financial Studies, 24,
Lecture 6: Non Normal Distributions
Lecture 6: Non Normal Distributions and their Uses in GARCH Modelling Prof. Massimo Guidolin 20192 Financial Econometrics Spring 2015 Overview Non-normalities in (standardized) residuals from asset return
More informationFinancial Econometrics Jeffrey R. Russell. Midterm 2014 Suggested Solutions. TA: B. B. Deng
Financial Econometrics Jeffrey R. Russell Midterm 2014 Suggested Solutions TA: B. B. Deng Unless otherwise stated, e t is iid N(0,s 2 ) 1. (12 points) Consider the three series y1, y2, y3, and y4. Match
More informationFinancial Econometrics
Financial Econometrics Volatility Gerald P. Dwyer Trinity College, Dublin January 2013 GPD (TCD) Volatility 01/13 1 / 37 Squared log returns for CRSP daily GPD (TCD) Volatility 01/13 2 / 37 Absolute value
More informationCross-Sectional Distribution of GARCH Coefficients across S&P 500 Constituents : Time-Variation over the Period
Cahier de recherche/working Paper 13-13 Cross-Sectional Distribution of GARCH Coefficients across S&P 500 Constituents : Time-Variation over the Period 2000-2012 David Ardia Lennart F. Hoogerheide Mai/May
More informationTHE INFORMATION CONTENT OF IMPLIED VOLATILITY IN AGRICULTURAL COMMODITY MARKETS. Pierre Giot 1
THE INFORMATION CONTENT OF IMPLIED VOLATILITY IN AGRICULTURAL COMMODITY MARKETS Pierre Giot 1 May 2002 Abstract In this paper we compare the incremental information content of lagged implied volatility
More informationVolatility Analysis of Nepalese Stock Market
The Journal of Nepalese Business Studies Vol. V No. 1 Dec. 008 Volatility Analysis of Nepalese Stock Market Surya Bahadur G.C. Abstract Modeling and forecasting volatility of capital markets has been important
More informationConditional Heteroscedasticity
1 Conditional Heteroscedasticity May 30, 2010 Junhui Qian 1 Introduction ARMA(p,q) models dictate that the conditional mean of a time series depends on past observations of the time series and the past
More informationFINITE SAMPLE DISTRIBUTIONS OF RISK-RETURN RATIOS
Available Online at ESci Journals Journal of Business and Finance ISSN: 305-185 (Online), 308-7714 (Print) http://www.escijournals.net/jbf FINITE SAMPLE DISTRIBUTIONS OF RISK-RETURN RATIOS Reza Habibi*
More informationThe Great Moderation Flattens Fat Tails: Disappearing Leptokurtosis
The Great Moderation Flattens Fat Tails: Disappearing Leptokurtosis WenShwo Fang Department of Economics Feng Chia University 100 WenHwa Road, Taichung, TAIWAN Stephen M. Miller* College of Business University
More informationPrerequisites for modeling price and return data series for the Bucharest Stock Exchange
Theoretical and Applied Economics Volume XX (2013), No. 11(588), pp. 117-126 Prerequisites for modeling price and return data series for the Bucharest Stock Exchange Andrei TINCA The Bucharest University
More informationGARCH Models. Instructor: G. William Schwert
APS 425 Fall 2015 GARCH Models Instructor: G. William Schwert 585-275-2470 schwert@schwert.ssb.rochester.edu Autocorrelated Heteroskedasticity Suppose you have regression residuals Mean = 0, not autocorrelated
More informationINFORMATION EFFICIENCY HYPOTHESIS THE FINANCIAL VOLATILITY IN THE CZECH REPUBLIC CASE
INFORMATION EFFICIENCY HYPOTHESIS THE FINANCIAL VOLATILITY IN THE CZECH REPUBLIC CASE Abstract Petr Makovský If there is any market which is said to be effective, this is the the FOREX market. Here we
More informationIndian Institute of Management Calcutta. Working Paper Series. WPS No. 797 March Implied Volatility and Predictability of GARCH Models
Indian Institute of Management Calcutta Working Paper Series WPS No. 797 March 2017 Implied Volatility and Predictability of GARCH Models Vivek Rajvanshi Assistant Professor, Indian Institute of Management
More informationStatistical Analysis of Data from the Stock Markets. UiO-STK4510 Autumn 2015
Statistical Analysis of Data from the Stock Markets UiO-STK4510 Autumn 2015 Sampling Conventions We observe the price process S of some stock (or stock index) at times ft i g i=0,...,n, we denote it by
More informationResearch Article The Volatility of the Index of Shanghai Stock Market Research Based on ARCH and Its Extended Forms
Discrete Dynamics in Nature and Society Volume 2009, Article ID 743685, 9 pages doi:10.1155/2009/743685 Research Article The Volatility of the Index of Shanghai Stock Market Research Based on ARCH and
More informationGlobal Journal of Finance and Banking Issues Vol. 5. No Manu Sharma & Rajnish Aggarwal PERFORMANCE ANALYSIS OF HEDGE FUND INDICES
PERFORMANCE ANALYSIS OF HEDGE FUND INDICES Dr. Manu Sharma 1 Panjab University, India E-mail: manumba2000@yahoo.com Rajnish Aggarwal 2 Panjab University, India Email: aggarwalrajnish@gmail.com Abstract
More informationMEASURING PORTFOLIO RISKS USING CONDITIONAL COPULA-AR-GARCH MODEL
MEASURING PORTFOLIO RISKS USING CONDITIONAL COPULA-AR-GARCH MODEL Isariya Suttakulpiboon MSc in Risk Management and Insurance Georgia State University, 30303 Atlanta, Georgia Email: suttakul.i@gmail.com,
More informationLecture 5a: ARCH Models
Lecture 5a: ARCH Models 1 2 Big Picture 1. We use ARMA model for the conditional mean 2. We use ARCH model for the conditional variance 3. ARMA and ARCH model can be used together to describe both conditional
More informationLecture 1: Empirical Properties of Returns
Lecture 1: Empirical Properties of Returns Econ 589 Eric Zivot Spring 2011 Updated: March 29, 2011 Daily CC Returns on MSFT -0.3 r(t) -0.2-0.1 0.1 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996
More informationBooth School of Business, University of Chicago Business 41202, Spring Quarter 2014, Mr. Ruey S. Tsay. Solutions to Midterm
Booth School of Business, University of Chicago Business 41202, Spring Quarter 2014, Mr. Ruey S. Tsay Solutions to Midterm Problem A: (30 pts) Answer briefly the following questions. Each question has
More informationExample 1 of econometric analysis: the Market Model
Example 1 of econometric analysis: the Market Model IGIDR, Bombay 14 November, 2008 The Market Model Investors want an equation predicting the return from investing in alternative securities. Return is
More informationRisk Spillovers of Financial Institutions
Risk Spillovers of Financial Institutions Tobias Adrian and Markus K. Brunnermeier Federal Reserve Bank of New York and Princeton University Risk Transfer Mechanisms and Financial Stability Basel, 29-30
More informationBooth School of Business, University of Chicago Business 41202, Spring Quarter 2012, Mr. Ruey S. Tsay. Solutions to Midterm
Booth School of Business, University of Chicago Business 41202, Spring Quarter 2012, Mr. Ruey S. Tsay Solutions to Midterm Problem A: (34 pts) Answer briefly the following questions. Each question has
More informationAssicurazioni Generali: An Option Pricing Case with NAGARCH
Assicurazioni Generali: An Option Pricing Case with NAGARCH Assicurazioni Generali: Business Snapshot Find our latest analyses and trade ideas on bsic.it Assicurazioni Generali SpA is an Italy-based insurance
More informationHedge Fund Volatility: It s Not What You Think It Is 1 By Clifford De Souza, Ph.D., and Suleyman Gokcan 2, Ph.D. Citigroup Alternative Investments
Disclaimer: This article appeared in the AIMA Journal (Sept 2004), which is published by The Alternative Investment 1 Hedge Fd Volatility: It s Not What You Think It Is 1 By Clifford De Souza, Ph.D., and
More informationSensex Realized Volatility Index (REALVOL)
Sensex Realized Volatility Index (REALVOL) Introduction Volatility modelling has traditionally relied on complex econometric procedures in order to accommodate the inherent latent character of volatility.
More informationFinancial Econometrics (FinMetrics04) Time-series Statistics Concepts Exploratory Data Analysis Testing for Normality Empirical VaR
Financial Econometrics (FinMetrics04) Time-series Statistics Concepts Exploratory Data Analysis Testing for Normality Empirical VaR Nelson Mark University of Notre Dame Fall 2017 September 11, 2017 Introduction
More informationCHAPTER II LITERATURE STUDY
CHAPTER II LITERATURE STUDY 2.1. Risk Management Monetary crisis that strike Indonesia during 1998 and 1999 has caused bad impact to numerous government s and commercial s bank. Most of those banks eventually
More information1. You are given the following information about a stationary AR(2) model:
Fall 2003 Society of Actuaries **BEGINNING OF EXAMINATION** 1. You are given the following information about a stationary AR(2) model: (i) ρ 1 = 05. (ii) ρ 2 = 01. Determine φ 2. (A) 0.2 (B) 0.1 (C) 0.4
More informationRETURNS AND VOLATILITY SPILLOVERS IN BRIC (BRAZIL, RUSSIA, INDIA, CHINA), EUROPE AND USA
RETURNS AND VOLATILITY SPILLOVERS IN BRIC (BRAZIL, RUSSIA, INDIA, CHINA), EUROPE AND USA Burhan F. Yavas, College of Business Administrations and Public Policy California State University Dominguez Hills
More informationFINANCIAL ECONOMETRICS AND EMPIRICAL FINANCE MODULE 2
MSc. Finance/CLEFIN 2017/2018 Edition FINANCIAL ECONOMETRICS AND EMPIRICAL FINANCE MODULE 2 Midterm Exam Solutions June 2018 Time Allowed: 1 hour and 15 minutes Please answer all the questions by writing
More informationA Robust Test for Normality
A Robust Test for Normality Liangjun Su Guanghua School of Management, Peking University Ye Chen Guanghua School of Management, Peking University Halbert White Department of Economics, UCSD March 11, 2006
More informationUniversal Properties of Financial Markets as a Consequence of Traders Behavior: an Analytical Solution
Universal Properties of Financial Markets as a Consequence of Traders Behavior: an Analytical Solution Simone Alfarano, Friedrich Wagner, and Thomas Lux Institut für Volkswirtschaftslehre der Christian
More informationPersistence Analysis of Hedge Fund Returns
Persistence Analysis of Hedge Fund Returns Serge Patrick Amvella, Iwan Meier, Nicolas Papageorgiou HEC Montréal November 0, 009 Abstract We use a Markov chain model to evaluate pure persistence in hedge
More informationVolatility Clustering of Fine Wine Prices assuming Different Distributions
Volatility Clustering of Fine Wine Prices assuming Different Distributions Cynthia Royal Tori, PhD Valdosta State University Langdale College of Business 1500 N. Patterson Street, Valdosta, GA USA 31698
More informationFinancial Risk Forecasting Chapter 9 Extreme Value Theory
Financial Risk Forecasting Chapter 9 Extreme Value Theory Jon Danielsson 2017 London School of Economics To accompany Financial Risk Forecasting www.financialriskforecasting.com Published by Wiley 2011
More informationCan Factor Timing Explain Hedge Fund Alpha?
Can Factor Timing Explain Hedge Fund Alpha? Hyuna Park Minnesota State University, Mankato * First Draft: June 12, 2009 This Version: December 23, 2010 Abstract Hedge funds are in a better position than
More information12. Conditional heteroscedastic models (ARCH) MA6622, Ernesto Mordecki, CityU, HK, 2006.
12. Conditional heteroscedastic models (ARCH) MA6622, Ernesto Mordecki, CityU, HK, 2006. References for this Lecture: Robert F. Engle. Autoregressive Conditional Heteroscedasticity with Estimates of Variance
More informationModeling Exchange Rate Volatility using APARCH Models
96 TUTA/IOE/PCU Journal of the Institute of Engineering, 2018, 14(1): 96-106 TUTA/IOE/PCU Printed in Nepal Carolyn Ogutu 1, Betuel Canhanga 2, Pitos Biganda 3 1 School of Mathematics, University of Nairobi,
More informationVladimir Spokoiny (joint with J.Polzehl) Varying coefficient GARCH versus local constant volatility modeling.
W e ie rstra ß -In stitu t fü r A n g e w a n d te A n a ly sis u n d S to c h a stik STATDEP 2005 Vladimir Spokoiny (joint with J.Polzehl) Varying coefficient GARCH versus local constant volatility modeling.
More informationThe Analysis of ICBC Stock Based on ARMA-GARCH Model
Volume 04 - Issue 08 August 2018 PP. 11-16 The Analysis of ICBC Stock Based on ARMA-GARCH Model Si-qin LIU 1 Hong-guo SUN 1* 1 (Department of Mathematics and Finance Hunan University of Humanities Science
More informationPerformance Dynamics of Hedge Fund Index Investing
Journal of Business and Economics, ISSN 2155-7950, USA November 2016, Volume 7, No. 11, pp. 1729-1742 DOI: 10.15341/jbe(2155-7950)/11.07.2016/001 Academic Star Publishing Company, 2016 http://www.academicstar.us
More informationBooth School of Business, University of Chicago Business 41202, Spring Quarter 2016, Mr. Ruey S. Tsay. Solutions to Midterm
Booth School of Business, University of Chicago Business 41202, Spring Quarter 2016, Mr. Ruey S. Tsay Solutions to Midterm Problem A: (30 pts) Answer briefly the following questions. Each question has
More informationFinancial Returns: Stylized Features and Statistical Models
Financial Returns: Stylized Features and Statistical Models Qiwei Yao Department of Statistics London School of Economics q.yao@lse.ac.uk p.1 Definitions of returns Empirical evidence: daily prices in
More informationModeling the volatility of FTSE All Share Index Returns
MPRA Munich Personal RePEc Archive Modeling the volatility of FTSE All Share Index Returns Bayraci, Selcuk University of Exeter, Yeditepe University 27. April 2007 Online at http://mpra.ub.uni-muenchen.de/28095/
More informationForecasting Stock Index Futures Price Volatility: Linear vs. Nonlinear Models
The Financial Review 37 (2002) 93--104 Forecasting Stock Index Futures Price Volatility: Linear vs. Nonlinear Models Mohammad Najand Old Dominion University Abstract The study examines the relative ability
More informationA Regime Switching model
Master Degree Project in Finance A Regime Switching model Applied to the OMXS30 and Nikkei 225 indices Ludvig Hjalmarsson Supervisor: Mattias Sundén Master Degree Project No. 2014:92 Graduate School Masters
More informationMODELLING OF INCOME AND WAGE DISTRIBUTION USING THE METHOD OF L-MOMENTS OF PARAMETER ESTIMATION
International Days of Statistics and Economics, Prague, September -3, MODELLING OF INCOME AND WAGE DISTRIBUTION USING THE METHOD OF L-MOMENTS OF PARAMETER ESTIMATION Diana Bílková Abstract Using L-moments
More informationAnalyzing Oil Futures with a Dynamic Nelson-Siegel Model
Analyzing Oil Futures with a Dynamic Nelson-Siegel Model NIELS STRANGE HANSEN & ASGER LUNDE DEPARTMENT OF ECONOMICS AND BUSINESS, BUSINESS AND SOCIAL SCIENCES, AARHUS UNIVERSITY AND CENTER FOR RESEARCH
More informationOptimal Portfolio Inputs: Various Methods
Optimal Portfolio Inputs: Various Methods Prepared by Kevin Pei for The Fund @ Sprott Abstract: In this document, I will model and back test our portfolio with various proposed models. It goes without
More informationAn Empirical Research on Chinese Stock Market Volatility Based. on Garch
Volume 04 - Issue 07 July 2018 PP. 15-23 An Empirical Research on Chinese Stock Market Volatility Based on Garch Ya Qian Zhu 1, Wen huili* 1 (Department of Mathematics and Finance, Hunan University of
More informationStatistical Inference and Methods
Department of Mathematics Imperial College London d.stephens@imperial.ac.uk http://stats.ma.ic.ac.uk/ das01/ 14th February 2006 Part VII Session 7: Volatility Modelling Session 7: Volatility Modelling
More information1 Volatility Definition and Estimation
1 Volatility Definition and Estimation 1.1 WHAT IS VOLATILITY? It is useful to start with an explanation of what volatility is, at least for the purpose of clarifying the scope of this book. Volatility
More informationBooth School of Business, University of Chicago Business 41202, Spring Quarter 2013, Mr. Ruey S. Tsay. Midterm
Booth School of Business, University of Chicago Business 41202, Spring Quarter 2013, Mr. Ruey S. Tsay Midterm ChicagoBooth Honor Code: I pledge my honor that I have not violated the Honor Code during this
More informationGARCH Models for Inflation Volatility in Oman
Rev. Integr. Bus. Econ. Res. Vol 2(2) 1 GARCH Models for Inflation Volatility in Oman Muhammad Idrees Ahmad Department of Mathematics and Statistics, College of Science, Sultan Qaboos Universty, Alkhod,
More informationFinancial Econometrics Jeffrey R. Russell Midterm 2014
Name: Financial Econometrics Jeffrey R. Russell Midterm 2014 You have 2 hours to complete the exam. Use can use a calculator and one side of an 8.5x11 cheat sheet. Try to fit all your work in the space
More informationBooth School of Business, University of Chicago Business 41202, Spring Quarter 2012, Mr. Ruey S. Tsay. Midterm
Booth School of Business, University of Chicago Business 41202, Spring Quarter 2012, Mr. Ruey S. Tsay Midterm ChicagoBooth Honor Code: I pledge my honor that I have not violated the Honor Code during this
More informationWindow Width Selection for L 2 Adjusted Quantile Regression
Window Width Selection for L 2 Adjusted Quantile Regression Yoonsuh Jung, The Ohio State University Steven N. MacEachern, The Ohio State University Yoonkyung Lee, The Ohio State University Technical Report
More informationVariance clustering. Two motivations, volatility clustering, and implied volatility
Variance modelling The simplest assumption for time series is that variance is constant. Unfortunately that assumption is often violated in actual data. In this lecture we look at the implications of time
More informationA market risk model for asymmetric distributed series of return
University of Wollongong Research Online University of Wollongong in Dubai - Papers University of Wollongong in Dubai 2012 A market risk model for asymmetric distributed series of return Kostas Giannopoulos
More informationKey Words: emerging markets, copulas, tail dependence, Value-at-Risk JEL Classification: C51, C52, C14, G17
RISK MANAGEMENT WITH TAIL COPULAS FOR EMERGING MARKET PORTFOLIOS Svetlana Borovkova Vrije Universiteit Amsterdam Faculty of Economics and Business Administration De Boelelaan 1105, 1081 HV Amsterdam, The
More informationMEMBER CONTRIBUTION. 20 years of VIX: Implications for Alternative Investment Strategies
MEMBER CONTRIBUTION 20 years of VIX: Implications for Alternative Investment Strategies Mikhail Munenzon, CFA, CAIA, PRM Director of Asset Allocation and Risk, The Observatory mikhail@247lookout.com Copyright
More informationJaime Frade Dr. Niu Interest rate modeling
Interest rate modeling Abstract In this paper, three models were used to forecast short term interest rates for the 3 month LIBOR. Each of the models, regression time series, GARCH, and Cox, Ingersoll,
More informationJohn Hull, Risk Management and Financial Institutions, 4th Edition
P1.T2. Quantitative Analysis John Hull, Risk Management and Financial Institutions, 4th Edition Bionic Turtle FRM Video Tutorials By David Harper, CFA FRM 1 Chapter 10: Volatility (Learning objectives)
More information**BEGINNING OF EXAMINATION** A random sample of five observations from a population is:
**BEGINNING OF EXAMINATION** 1. You are given: (i) A random sample of five observations from a population is: 0.2 0.7 0.9 1.1 1.3 (ii) You use the Kolmogorov-Smirnov test for testing the null hypothesis,
More informationApplication of Conditional Autoregressive Value at Risk Model to Kenyan Stocks: A Comparative Study
American Journal of Theoretical and Applied Statistics 2017; 6(3): 150-155 http://www.sciencepublishinggroup.com/j/ajtas doi: 10.11648/j.ajtas.20170603.13 ISSN: 2326-8999 (Print); ISSN: 2326-9006 (Online)
More informationRegime-dependent Characteristics of KOSPI Return
Communications for Statistical Applications and Methods 014, Vol. 1, No. 6, 501 51 DOI: http://dx.doi.org/10.5351/csam.014.1.6.501 Print ISSN 87-7843 / Online ISSN 383-4757 Regime-dependent Characteristics
More informationAmath 546/Econ 589 Univariate GARCH Models: Advanced Topics
Amath 546/Econ 589 Univariate GARCH Models: Advanced Topics Eric Zivot April 29, 2013 Lecture Outline The Leverage Effect Asymmetric GARCH Models Forecasts from Asymmetric GARCH Models GARCH Models with
More informationAssessing Regime Switching Equity Return Models
Assessing Regime Switching Equity Return Models R. Keith Freeland, ASA, Ph.D. Mary R. Hardy, FSA, FIA, CERA, Ph.D. Matthew Till Copyright 2009 by the Society of Actuaries. All rights reserved by the Society
More informationAre Market Neutral Hedge Funds Really Market Neutral?
Are Market Neutral Hedge Funds Really Market Neutral? Andrew Patton London School of Economics June 2005 1 Background The hedge fund industry has grown from about $50 billion in 1990 to $1 trillion in
More informationHedge Funds performance during the recent financial crisis. Master Thesis
Hedge Funds performance during the recent financial crisis Master Thesis Ioannis Politidis ANR:146310 Supervisor: R.G.P Frehen 26 th November 2013 Tilburg University Tilburg School of Economics and Management
More informationThe Economic and Social BOOTSTRAPPING Review, Vol. 31, No. THE 4, R/S October, STATISTIC 2000, pp
The Economic and Social BOOTSTRAPPING Review, Vol. 31, No. THE 4, R/S October, STATISTIC 2000, pp. 351-359 351 Bootstrapping the Small Sample Critical Values of the Rescaled Range Statistic* MARWAN IZZELDIN
More informationFinancial Data Analysis, WS08/09. Roman Liesenfeld, University of Kiel 1
Financial Data Analysis, WS08/09. Roman Liesenfeld, University of Kiel 1 Data sets used in the following sections can be downloaded from http://faculty.chicagogsb.edu/ruey.tsay/teaching/fts/ Exercise Sheet
More informationChapter 4 Level of Volatility in the Indian Stock Market
Chapter 4 Level of Volatility in the Indian Stock Market Measurement of volatility is an important issue in financial econometrics. The main reason for the prominent role that volatility plays in financial
More informationARCH and GARCH models
ARCH and GARCH models Fulvio Corsi SNS Pisa 5 Dic 2011 Fulvio Corsi ARCH and () GARCH models SNS Pisa 5 Dic 2011 1 / 21 Asset prices S&P 500 index from 1982 to 2009 1600 1400 1200 1000 800 600 400 200
More informationValue at risk might underestimate risk when risk bites. Just bootstrap it!
23 September 215 by Zhili Cao Research & Investment Strategy at risk might underestimate risk when risk bites. Just bootstrap it! Key points at Risk (VaR) is one of the most widely used statistical tools
More informationForecasting Volatility of USD/MUR Exchange Rate using a GARCH (1,1) model with GED and Student s-t errors
UNIVERSITY OF MAURITIUS RESEARCH JOURNAL Volume 17 2011 University of Mauritius, Réduit, Mauritius Research Week 2009/2010 Forecasting Volatility of USD/MUR Exchange Rate using a GARCH (1,1) model with
More informationEmpirical Analysis of Stock Return Volatility with Regime Change: The Case of Vietnam Stock Market
7/8/1 1 Empirical Analysis of Stock Return Volatility with Regime Change: The Case of Vietnam Stock Market Vietnam Development Forum Tokyo Presentation By Vuong Thanh Long Dept. of Economic Development
More informationIntroduction to Computational Finance and Financial Econometrics Descriptive Statistics
You can t see this text! Introduction to Computational Finance and Financial Econometrics Descriptive Statistics Eric Zivot Summer 2015 Eric Zivot (Copyright 2015) Descriptive Statistics 1 / 28 Outline
More informationStock Price Volatility in European & Indian Capital Market: Post-Finance Crisis
International Review of Business and Finance ISSN 0976-5891 Volume 9, Number 1 (2017), pp. 45-55 Research India Publications http://www.ripublication.com Stock Price Volatility in European & Indian Capital
More informationSome Simple Stochastic Models for Analyzing Investment Guarantees p. 1/36
Some Simple Stochastic Models for Analyzing Investment Guarantees Wai-Sum Chan Department of Statistics & Actuarial Science The University of Hong Kong Some Simple Stochastic Models for Analyzing Investment
More informationHigh-Frequency Data Analysis and Market Microstructure [Tsay (2005), chapter 5]
1 High-Frequency Data Analysis and Market Microstructure [Tsay (2005), chapter 5] High-frequency data have some unique characteristics that do not appear in lower frequencies. At this class we have: Nonsynchronous
More informationVolume Author/Editor: Joseph G. Haubrich and Andrew W. Lo, editors. Volume Publisher: University of Chicago Press
This PDF is a selection from a published volume from the National Bureau of Economic Research Volume Title: Quantifying Systemic Risk Volume Author/Editor: Joseph G. Haubrich and Andrew W. Lo, editors
More informationEconometric Models for the Analysis of Financial Portfolios
Econometric Models for the Analysis of Financial Portfolios Professor Gabriela Victoria ANGHELACHE, Ph.D. Academy of Economic Studies Bucharest Professor Constantin ANGHELACHE, Ph.D. Artifex University
More informationModelling Joint Distribution of Returns. Dr. Sawsan Hilal space
Modelling Joint Distribution of Returns Dr. Sawsan Hilal space Maths Department - University of Bahrain space October 2011 REWARD Asset Allocation Problem PORTFOLIO w 1 w 2 w 3 ASSET 1 ASSET 2 R 1 R 2
More information2.4 STATISTICAL FOUNDATIONS
2.4 STATISTICAL FOUNDATIONS Characteristics of Return Distributions Moments of Return Distribution Correlation Standard Deviation & Variance Test for Normality of Distributions Time Series Return Volatility
More informationOccasional Paper. Dynamic Methods for Analyzing Hedge-Fund Performance: A Note Using Texas Energy-Related Funds. Jiaqi Chen and Michael L.
DALLASFED Occasional Paper Dynamic Methods for Analyzing Hedge-Fund Performance: A Note Using Texas Energy-Related Funds Jiaqi Chen and Michael L. Tindall Federal Reserve Bank of Dallas Financial Industry
More informationRisk Management and Time Series
IEOR E4602: Quantitative Risk Management Spring 2016 c 2016 by Martin Haugh Risk Management and Time Series Time series models are often employed in risk management applications. They can be used to estimate
More informationInflation and inflation uncertainty in Argentina,
U.S. Department of the Treasury From the SelectedWorks of John Thornton March, 2008 Inflation and inflation uncertainty in Argentina, 1810 2005 John Thornton Available at: https://works.bepress.com/john_thornton/10/
More informationHypothesis Tests: One Sample Mean Cal State Northridge Ψ320 Andrew Ainsworth PhD
Hypothesis Tests: One Sample Mean Cal State Northridge Ψ320 Andrew Ainsworth PhD MAJOR POINTS Sampling distribution of the mean revisited Testing hypotheses: sigma known An example Testing hypotheses:
More informationValue at Risk with Stable Distributions
Value at Risk with Stable Distributions Tecnológico de Monterrey, Guadalajara Ramona Serrano B Introduction The core activity of financial institutions is risk management. Calculate capital reserves given
More informationOil Price Effects on Exchange Rate and Price Level: The Case of South Korea
Oil Price Effects on Exchange Rate and Price Level: The Case of South Korea Mirzosaid SULTONOV 東北公益文科大学総合研究論集第 34 号抜刷 2018 年 7 月 30 日発行 研究論文 Oil Price Effects on Exchange Rate and Price Level: The Case
More informationFinancial Econometrics
Financial Econometrics Introduction to Financial Econometrics Gerald P. Dwyer Trinity College, Dublin January 2016 Outline 1 Set Notation Notation for returns 2 Summary statistics for distribution of data
More informationAsset Allocation Model with Tail Risk Parity
Proceedings of the Asia Pacific Industrial Engineering & Management Systems Conference 2017 Asset Allocation Model with Tail Risk Parity Hirotaka Kato Graduate School of Science and Technology Keio University,
More informationExtend the ideas of Kan and Zhou paper on Optimal Portfolio Construction under parameter uncertainty
Extend the ideas of Kan and Zhou paper on Optimal Portfolio Construction under parameter uncertainty George Photiou Lincoln College University of Oxford A dissertation submitted in partial fulfilment for
More informationINTERNATIONAL JOURNAL OF ADVANCED RESEARCH IN ENGINEERING AND TECHNOLOGY (IJARET)
INTERNATIONAL JOURNAL OF ADVANCED RESEARCH IN ENGINEERING AND TECHNOLOGY (IJARET) ISSN 0976-6480 (Print) ISSN 0976-6499 (Online) Volume 5, Issue 3, March (204), pp. 73-82 IAEME: www.iaeme.com/ijaret.asp
More informationAssessing Regime Switching Equity Return Models
Assessing Regime Switching Equity Return Models R. Keith Freeland Mary R Hardy Matthew Till January 28, 2009 In this paper we examine time series model selection and assessment based on residuals, with
More informationANALYSIS OF STOCHASTIC PROCESSES: CASE OF AUTOCORRELATION OF EXCHANGE RATES
Abstract ANALYSIS OF STOCHASTIC PROCESSES: CASE OF AUTOCORRELATION OF EXCHANGE RATES Mimoun BENZAOUAGH Ecole Supérieure de Technologie, Université IBN ZOHR Agadir, Maroc The present work consists of explaining
More informationAn Empirical Analysis of Effect on Copper Futures Yield. Based on GARCH
An Empirical Analysis of Effect on Copper Futures Yield Based on GARCH Feng Li 1, Ping Xiao 2 * 1 (School of Hunan University of Humanities, Science and Technology, Hunan 417000, China) 2 (School of Hunan
More informationWashington University Fall Economics 487
Washington University Fall 2009 Department of Economics James Morley Economics 487 Project Proposal due Tuesday 11/10 Final Project due Wednesday 12/9 (by 5:00pm) (20% penalty per day if the project is
More information